Re: monic epics

[David Karapetyan <dkarapetyan@ucdavis.edu>]

Michael Barr wrote:
> Category theory is too abstract for any such statement to be true (or
> even make sense).  For example, in the category denoted . ---> . (with
> two objects and one non-identity map), that map is monic and epic for
> want of any test maps.  More concretely, the inclusion of Z into R is
> both in the category of commtative rings.  In fact the following a
> characterization of monic/epics in commutative rings is this: a
> subring R \inc S is epic iff every element of of S can be written s =
> vAw where for some n, v is an n-dimensional row vector, w is an
> n-dimensional column vector and A is an n x n matrix of elements of S
> such that the entries of A, vA, and Aw all belong to R.  In general,
> very little can be said about monic/epics.
>
ok i got it. in all the examples given the subobjects given by the
monics are "generators" for the object, where by "generators" i mean the
elements of the subobject in some way determine the elements of the
bigger object. so how about this then: any time we have the situation
described above the monic arrow will also be epic.